Ternary is the base-3 numeral system in which only the digits 0, 1, and 2 are used.
Ternary numbers arise in a number of problems in mathematics, including some problems
of weighing. However, according to Knuth (1998), "no
substantial application of balanced ternary notation has been made" (balanced
ternary uses digits 
, 0, and 1 instead of 0, 1, and 2).
The illustration above shows a graphical representation of the integers from 0 to 25 in ternary, and the following table gives the ternary equivalents of the first few decimal integers. The concatenation of the ternary digits of the consecutive numbers 0, 1, 2, 3, ... gives (0), (1), (2), (1, 0), (1, 1), (1, 2), (2, 0), ... (OEIS A054635).
| 1 | 1 | 11 | 102 | 21 | 210 |
| 2 | 2 | 12 | 110 | 22 | 211 |
| 3 | 10 | 13 | 111 | 23 | 212 |
| 4 | 11 | 14 | 112 | 24 | 220 |
| 5 | 12 | 15 | 120 | 25 | 221 |
| 6 | 20 | 16 | 121 | 26 | 222 |
| 7 | 21 | 17 | 122 | 27 | 1000 |
| 8 | 22 | 18 | 200 | 28 | 1001 |
| 9 | 100 | 19 | 201 | 29 | 1002 |
| 10 | 101 | 20 | 202 | 30 | 1010 |
Ternary digits have the following multiplication table.
 | 0 | 1 | 2 |
| 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 |
| 2 | 0 | 2 | 11 |
For a three-state totalistic cellular automaton with a neighborhood of three cells, a ternary representation uniquely
identifies its rule. The three colors (white, gray, and black) correspond to the
three numbers 0, 1, and 2 (Wolfram 2002, pp. 60-70
and 886).
For example, the ternary digits 
lead to the code 600 totalistic
cellular automaton.
Every even number represented in ternary has an even number (possibly 0) of 1s. This is true since a
nonnegative integer is congruent modulo 
to its digit
sum in base 
. In the case 
, there is only one digit (1) which
is not a multiple of 
, so all we have to do is "cast out twos" and count
the number of 1s in the base-3 representation.
The following table gives 
for 
, 2, ... in ternary.
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
 |  |  |
(7)
|
N. J. A. Sloane conjectured that for any integer 
, 
always has a 0 in its ternary expansion (Sloane 1973, p. 98;
Vardi 1991, p. 28). Known positive integer
values of 
such that 
lacks
a 0 begin with 1, 2, 3, 4, 15, ... (OEIS A102483),
with no others up to 
(E. W. Weisstein, Apr. 8, 2006). The positions (counting from 1 at
the least significant ternary digit, with 0 denoting absence)
of the first 0 digit in 
, 
,
..., are 0, 0, 0, 0, 3, 2, 2, 4, 4, 5, 4, 2, 2, 4, 0, 3, 4, ... (OEIS A117970).
It is conjectured that, for positive integers 
, 
has a 1 in its ternary expansion except when 
, 3, or 9. This is equivalent to the conjecture
about the occurrence of a 2 in 
(Saye 2022, pp. 1-2). No further exceptions were
found up to 
(E. W. Weisstein, Apr. 8, 2006).
Erdős and Graham (1980) conjectured that no power of 2, 
, for integers 
is a sum
of distinct powers of 3. This is equivalent to the requirement
that the ternary expansion of 
always contains a 2 for 
.
For positive integers 
, Vardi (1991) verified that the only values of 
for which 
has no 2 in its ternary expansion are 2 and 8 up to 
.
The positions (counting from 1 at the least significant ternary digit,
with 0 denoting absence) of the first 2 digit in 
, 
,
..., are 1, 0, 1, 2, 1, 4, 1, 0, 1, 2, 1, 3, 1, 3, ... (OEIS A117971).
in Base Three."